Multiple choice

The number of integral values of m for which the equation (1 + m2)x2 - 2(1 + 3m)x + (1 + 8m) = 0 has no real root is

  1. infinitely many

  2. 2

  3. 3

  4. 1
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A Correct answer
Explanation

For the quadratic equation to have no real roots, the discriminant D must be less than 0. D = [-2(1+3m)]^2 - 4(1+m^2)(1+8m) < 0. Expanding this leads to 4(1 + 6m + 9m^2) - 4(1 + 8m + m^2 + 8m^3) < 0, which simplifies to 4(1 + 6m + 9m^2 - 1 - 8m - m^2 - 8m^3) < 0, or -8m^3 + 8m^2 - 2m < 0. This inequality holds for many values of m.

AI explanation

For the quadratic equation to have no real roots, the discriminant must be less than zero, so D = b^2 - 4ac < 0. Substituting the coefficients gives (-2(1 + 3m))^2 - 4(1 + m2)(1 + 8m) < 0, which simplifies to 4(1 + 6m + 9m2) - 4(1 + 8m + m2 + 8m3) < 0. Dividing by 4 and canceling terms yields 8m3 - m2 - 2m > 0, or m(8m2 - m - 2) > 0. This cubic inequality holds true for infinitely many integral values of m, such as all integers greater than 1. The number of integral values is infinitely many.

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